- Smooth manifolds. Vector fields. Differential forms. Distributions. Relation to commutative algebra.
- Geometry of ordinary equations. Symmetries. Applications of symmetries to solving ODEs. Lie-Bianchi theorem on integration of ODEs by quadratures.
- Contact geometry and the theory of first-order equations. Relation to symplectic geometry and Hamiltonian mechanics.
- Finite jets of submanifolds. Cartan distribution. Integral manifolds. Lie-Baeklund theorem.
- Differential equations as geometric objects. Theory of symmetries. Application of symmetries (invariant solutions, reproduction of solutions, factorization). Examples. External and internal geometry of equations.
- Infinite jets. Algebraic formalism. Prolongation of differential equations. Higher symmetries and their computation. Examples. Computer methods for finding symmetries.
- Conservation laws and their computation. Noether theorem. Hamiltonian structures. Examples.
- Coverings over differential equations and nonlocal symmetries. Baeklund transformations.

Questions and suggestions should go to Jet Nestruev, jet @ diffiety.ac.ru.